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Problem 6 You have been commissioned by Oxford ci...

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Problem 6 You have been commissioned by Oxford city government to study how the addition of 1,000 condominium units around the edge of town over the last several years might impact the city's transportation infrastructure, levels of trac, and air quality. You decide to begin by estimating the number of additional motor vehicles brought to the area per new condominium. In a survey of the occupants, a simple random sample of size 100 yields the information that the average number of motor vehicles per unit is 1.6, with a sample standard deviation of 0.8. A) Calculate the estimated standard error of X n=100, which we will label Sx. B) Calculate a 50% con dence interval, 90% con dence interval, and a 99% con dence interval for the population mean. Explain how the formulas you are using di er. What happens to the interval as you increase the level of con dence? C) You are now being interviewed by The Oxford Eagle, the local newspaper, about your initial survey results. i) Give the newspaper a brief explanation about your results in Part B. Be rigorous, but use easy-to- understand language (i.e., plain English) about what the con dence interval means. ii) Popular media usually include with surveys like yours the margin of error." The margin of error is usually operationalized as the radius (or half-width) of a 90% con dence interval around the point estimate, although its exact usage varies from survey to survey. Provide the margin of error for your estimate based on your 90% con dence interval from Part B. 2 D) Oxford has hired another graduate student to do a similar analysis. He is using X Sx to calculate his con dence interval. What level of con dence is he using? E) The graduate student in Part D also computes a 95% con dence interval, which is (1.44, 1.76). However, he did not take POL 551, and it shows. He o ers the following explanations of this interval to local media. Explain what is wrong with each interpretation. i) If random samples of 100 condominiums were repeatedly selected, then 95% of the time the sample mean number of motor vehicles will fall between 1.44 and 1.76." ii) Ninety- ve percent of the number of motor vehicles for condominiums in the new developments are between 1.44 and 1.76. (In addition to the technical reason this interpretation is wrong, there is also an common sense reason)." iii) We can be 95% con dent that the sample mean is between 1.44 and 1.76." iv) If we repeatedly sampled the entire population of 1,000 condominiums, then 95% of the time the population mean would be between 1.44 and 1.76." F) Part B showed how the con dence level changes the width of your con dence interval, but other factors might shape the interval too. Consider the following: i) What if instead of a sample standard deviation of 0.8 you found a sample standard deviation of 0.4. What would the 90% con dence interval be in this case? How does it compare to the 90% interval from Part B? ii) What if instead of a sample mean of 1.6 you found a sample mean of 1 (still with a standard deviation of 0.8). What would the 90% con dence interval be in this case? How does it compare to the 90% interval from Part B? iii) What if instead of a sampling 100 condominiums, you sampled 900 and still had a sample mean of 1.6 and standard deviation of 0.8. What would the 90% con dence interval be in this case? How does it compare to the 90% interval from Part B? iv) Summarize in laymen's terms what you have just demonstrated about whether and how con dence level, sample standard deviation, sample mean, and sample size all a ect the interval, and be sure to explain why this makes sense. G) After estimating the number of cars in the new development, you wonder about how these cars might a ect air quality or gasoline supply in the Oxford area. As a rst step toward examining these questions, you decide to check the fuel eciency of motor vehicles in the new developments. Unfortunately, you are short on time because you have a POL 551 problem set due. You only have time to select a simple random 3 sample of 9 motor vehicles from the condominiums. They have a sample mean of 25.5 miles per gallon (mpg) and a sample standard deviation of 4.5 mpg. What is the 95% con dence interval? 4

 

Paper#11652 | Written in 18-Jul-2015

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